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A357457 Number of partitions of n into two or more distinct odd parts.

%I A357457 #14 Jul 25 2025 12:13:58
%S A357457 0,0,0,0,1,0,1,0,2,1,2,1,3,2,3,3,5,4,5,5,7,7,8,8,11,11,12,13,16,16,18,
%T A357457 19,23,24,26,28,33,34,37,40,46,48,52,56,63,67,72,77,87,92,98,106,117,
%U A357457 124,133,143,157,167,178,191,209,222,236,254,276,293,312,334
%N A357457 Number of partitions of n into two or more distinct odd parts.
%H A357457 Alois P. Heinz, <a href="/A357457/b357457.txt">Table of n, a(n) for n = 0..10000</a>
%H A357457 Harmandeep Kaur and Muhammad Asif Rana, <a href="https://arxiv.org/abs/2506.11447">Partitions with unique largest part and their generating functions</a>, arXiv:2506.11447 [math.CO], 2025. See p. 6.
%F A357457 a(n) = A000700(n) - A000035(n) for n > 0.
%p A357457 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(n>i^2, 0,
%p A357457       b(n, i-1)+`if`(i*2-1>n, 0, b(n-(i*2-1), i-1))))
%p A357457     end:
%p A357457 a:= n-> b(n, iquo(n+1, 2))-irem(max(1, n), 2):
%p A357457 seq(a(n), n=0..67);  # _Alois P. Heinz_, Jul 25 2025
%t A357457 Monitor[Table[Count[IntegerPartitions[n], _?(And[Length[#] > 1, DuplicateFreeQ[#], AllTrue[#, OddQ]] &)], {n, 0, 67}], n] (* _Michael De Vlieger_, Jun 23 2025 *)
%Y A357457 Cf. A000035, A000700, A357456.
%K A357457 nonn
%O A357457 0,9
%A A357457 _Ilya Gutkovskiy_, Sep 29 2022